The authors are grateful to Giuseppe Moscarini, Ricardo Reis, and Michael Woodford for useful discussions and comments. They also thank seminar participants at Columbia University, the Board of Governors of the Federal Reserve, and New York University and conference participants at the New York Area Monetary Policy Workshop. The views expressed here are the authors’ and not necessarily those of the Federal Reserve Bank of Atlanta or the Federal Reserve System. Any remaining errors are the authors’ responsibility. Please address questions regarding content to Pierpaolo Benigno, LUISS Guido Carli and Einaudi Institute of Economics and Finance, Viale Romania 32, 00197 Roma, Italy, pbenigno@luiss.it, or Anastasios Karantounias, Research Department, Federal Reserve Bank of Atlanta, 1000 Peachtree Street NE, Atlanta, GA 30309, anastasios.karantounias@atl.frb.org. Federal Reserve Bank of Atlanta working papers, including revised versions, are available on the Atlanta Fed’s website at www.frbatlanta.org. Click “Publications” and then “Working Papers.” To receive e-mail notifications about new papers, use frbatlanta.org/forms/subscribe.

FEDERAL RESERVE BANK of ATLANTA WORKING PAPER SERIES

Overconfidence, Subjective Perception, and Pricing Behavior Pierpaolo Benigno and Anastasios G. Karantounias Working Paper 2017-14 November 2017 Abstract: We study the implications of overconfidence for price setting in a monopolistic competition setup with incomplete information. Our price-setters overestimate their abilities to infer aggregate shocks from private signals. The fraction of uninformed firms is endogenous; firms can obtain information by paying a fixed cost. We find two results: (1) overconfident firms are less inclined to acquire information, and (2) prices might exhibit excess volatility driven by nonfundamental noise. We explore the empirical predictions of our model for idiosyncratic price volatility. JEL classification: D4;D8;E3 Key words: overconfidence, imperfect common knowledge, information acquisition, inflation volatility

1 Introduction

This paper studies the implications of the well-documented psychological bias of overconfi-

dence on the price-setting behavior of firms in a monopolistically competitive market with

incomplete information. In our model, firms receive a private signal about an aggregate

shock that influences marginal costs. They can acquire better information by paying a fixed

cost. Our price-setters are overconfident in their signals, i.e. they overestimate the precision

of their private information.

There are two important conclusions that we draw from this framework: first, overcon-

fidence implies that a large fraction of firms is going to optimally stay uninformed; second,

overconfident firms set prices that may be excessively volatile, since they are driven by under-

estimated noise. This volatility at the individual level is coupled with an aggregate inflation

rate that is smooth and persistent. So we provide a unified “noise” explanation of prices at

both the idiosyncratic and the aggregate level.

There is a vast psychological literature on overconfidence. The term has borne differ-

ent interpretations depending on the particular study but here we focus on the concept of

overprecision.1 The concept of overconfidence is rather new in macro models but has been

influential in the finance literature. For example, Barber and Odean (2001) analyze the im-

plications of overconfident male investors for excessive stock trading whereas Daniel et al.

(1998) and Daniel et al. (2001) focus on the effects of investor’s overconfidence on market

overreaction and asset pricing.

Overconfidence has not been found to be important only in experimental studies or in se-

tups where monetary loss is inconsequential: Oberlechner and Osler (2009) and Oberlechner

and Osler (2012) provide empirical evidence that overconfidence, in the sense of underes-

timation of uncertainty, is ubiquitous in currency markets and that overconfident traders

actually survive in the long-run. In a similar vein, Burnside et al. (2011) associate investor’s

overconfidence to the forward premium puzzle.

The prevalence of overconfidence has led to work that generates endogenously optimistic

biases in beliefs, by taking into account the utility benefits of good outcomes. A prominent

example is the theory of optimal expectations of Brunnermeier and Parker (2005). Another

example is the work of Ortoleva and Snowberg (2015), who generate overconfidence as a

consequence of correlational neglect and explore the implications for political ideology. For

our own purposes, we take overconfidence as given but explore how it interacts with fun-

damental and strategic uncertainty in our monopolistic competition setup. In the baseline

version of our economy, overconfidence can deter agents from obtaining information, since

1See the related literature section for details and for an overview of the use of overconfidence in variousfields of economics.

1

they rely more on their subjective perception of the world, but can also increase the incen-

tives to obtain information by increasing the implicit cost of staying uninformed through the

mechanism of higher-order beliefs.

In a dynamic extension of our economy, the presence of strategic complementarities and

the interaction with higher-order beliefs generates persistent responses of output to nominal

shocks, as in the imperfect common knowledge setup of Woodford (2002), where all agents are

uninformed. The persistence at the aggregate level, together with the excess volatility of over-

confident uninformed price-setters at the idiosyncratic level, leads to interesting empirical

predictions. To flesh them out, we explore how much idiosyncratic price volatility can be

generated by overconfidence.

Using micro data on consumer prices for the U.S. economy, Klenow and Kryvtsov (2008)

have documented that prices change frequently with an average absolute size of 8.5% on a

monthly basis and that these frequent movements are equally likely to be positive or negative

in sign. In contrast, aggregate inflation averages just at 0.6% over an horizon of 3 months. We

find that with rational price setters, our model implies that the average absolute size of price

changes is of the order of only 2%. For a reasonable degree of overconfidence documented in

experimental studies, we find that the average absolute size of price changes is of the order of

4− 5%. This is quite large for a model that does not rely at all on idiosyncratic fundamental

shocks like idiosyncratic productivity or demand shocks.

2 Related literature

Our setup is based on the imperfect common knowledge framework of Woodford (2002).

Lorenzoni (2009) provides a theory of demand shocks in setups with incomplete information,

Melosi (2014) devises methods for the structural estimation of these models and Adam (2007)

considers optimal monetary policy.

More generally, the imperfect common knowledge setup shares similarities with the ratio-

nal inattention theory of Sims (2003) in its emphasis on the difficulty of processing informa-

tion. Moscarini (2004) analyzes in a continuous time model the optimal sampling frequency

of noisy information under information-processing constraints and shows that it can also gen-

erate inertia. Morris and Shin (2006) have emphasized that in models with forward-looking

expectations even the existence of a small fraction of uninformed agents about the future

path of fundamentals can generate persistence in the price behavior.

The option to obtain information resembles Reis (2006), who builds a model where pro-

ducers decide when to acquire information but do not receive any private signals. The

contribution of Hellwig and Veldkamp (2009) focuses on the strategic complementarities in

2

information acquisition and their implications for multiple equilibria in setups without over-

confidence.

The literature in psychology is nicely summarized by Moore and Healy (2008). They

provide an extensive survey of the studies that document overconfidence and differentiate

between overestimation (of one’s abilities), overplacement (relative to others), and overpreci-

sion, which is the type of overconfidence we focus on. An example of overstimation is Clayson

(2005), who shows that students overestimate their performance in exams. A prominent ex-

ample of overplacement is the study of Svenson (1981): in a simultaneous study of US and

Swedish drivers, 88% of the US group and 77% of the Swedish group asked believed that

they are safer drivers than the median. Another example is Guthrie et al. (2001): 90% of 168

federal magistrate judges thought that they are above average as far as their reversal rate

on appeal is concerned. Regarding overprecision, studies like Soll and Klayman (2004) have

shown that subjective confidence intervals are systematically too narrow given the accuracy

of one’s information. Another classic reference is Lichtenstein et al. (1982).

In finance, other influential studies beyond the ones mentioned in the introduction are

Malmendier and Tate (2005), who devise methods of measuring CEO overconfidence based

on corporate investment and Scheinkman and Xiong (2003), who explore the potential of

overconfidence to generate speculative bubbles in a dynamic setup. Daniel et al. (2002)

consider the policy implications of overconfidence. The survey of Daniel and Hirshleifer

(2015) provides additional references.

Various studies in experimental economics have explored the interactions of overconfidence

and excess entry, see for example Camerer and Lovallo (1999). Furthermore, in micro theory

there are models on the effects of overconfidence on performance as in Compte and Postlewaite

(2004) and on the optimal menu of wage contracts as in Fan and Moscarini (2005). In

macroeconomics, Caliendo and Huang (2008) analyze the implications of overconfidence for

consumption and savings problems.

3 Economy

In this section, we present a partial-equilibrium model of price-setting behavior in which firms

have full information on the structure, parameters and variables of interest.2 We consider a

continuum of firms indexed by i on the unit interval [0, 1]. Each firm produces a good that is

differentiated in the preferences of consumers. We do not explicitly model neither consumer

preferences nor their optimization problem. We just assume what is needed to characterize

2The model is similar to the one used in Ball and Romer (1989), Ball and Romer (1991) and Blanchardand Kiyotaki (1987).

3

the price-setting problem of firms. Firms are profit maximizers and set their prices in a

monopolistic-competitive market. The problem of a generic firm j is to choose the price of

its product P (j) to maximize real profits given by

P (j)

PY (j)− W

PL(j) (1)

where Y (j) is the demand of good j given by

Y (j) =

(P (j)

P

)−εY (2)

that depends on the relative price of the good j with respect to the general price index P

given by

P =

[∫ 1

0

P (i)1−εdi

] 11−ε

(3)

and on aggregate production Y . The parameter ε (with ε > 1) denotes the elasticity of

substitution across differentiated goods in consumer preferences. Firms use labor L(j) to

produce goods through the production function Y (j) = AL(j), where A is a productivity

shock common to all firms; W is the nominal wage paid for one unit of labor in the labor

market.

In this market (not modeled here) we assume that the labor-supply schedule implies the

following relation between real wage and aggregate production

W

P= Y η (4)

with η > 0.3 We assume the existence of a monetary authority that has a perfect control on

the level of nominal spending in the economy. It follows that

M = PY (5)

where M , which may be labeled as money supply, is indeed controlled by the monetary

authority.

3This labor-supply schedule can be derived from the optimizing-behavior of households in a general-equilibrium model. In particular, η would be a combination of the risk-aversion coefficient in consumerpreferences and of the Frisch elasticity of substitution of labor supply, or in case of local labor market of εas well. Assuming a more general labor-supply schedule does not change the subsequent analysis.

4

We can substitute (2), (4) and (5) into (1) to define a profit function of firm j as

π(P (j), P, θ) ≡[P (j)

P− 1

A

(M

P

)η](P (j)

P

)−ε(M

P

). (6)

Profits of firm j are a function of the action (in game-theoretic sense) of firm j, P (j), of the

actions of all other firms synthesized by the index P and the vector θ ≡ (A,M). Firm j is of

measure zero with respect to the aggregate, so its pricing decision does not affect the general

price index P .

3.1 Full information

In the full information economy firm j chooses price P (j) to maximize profits (6). Let P †(j)

denote the profit-maximizing price, given by

P †(j) =ε

ε− 1

W

A=

ε

ε− 1

P

A

(M

P

)η(7)

which is just the familiar markup rule over nominal marginal cost. Since the right-hand

side of (7) is independent of j, all firms set the same price, P †(j) = P . This implies a full

information output equal to Y ∗ = ( Aε/(ε−1)

)1η . The full information price level is a function of

the exogenous money supply and equilibrium output,

P ∗ =M

Y ∗=

(ε

ε− 1

1

A

) 1η

M.

P ∗ is function solely of the aggregate shock θ and will be a useful statistic in the next section.

For later use, we can use the definition of P ∗ to write (7) as

P †(j) = P 1−ηP ∗η. (8)

This rewriting shows that the parameter η determines whether price-setting decisions are

strategic complements (the case 0 < η < 1) or strategic substitutes (η > 1). In the strategic-

complements case, there is a positive elasticity between the individual optimal price and

the aggregate price level. The elasticity is negative when pricing decisions are strategic

substitutes.

5

3.2 Incomplete information

Our incomplete information setup is based on two assumptions. First, firms do not know the

realization of the aggregate shock θ. Second, following the lead of Woodford (2002) and the

imperfect common knowledge literature, we assume that the general price level P is not in

the information set of price-setters.

In particular, each firm receives a private signal about θ, that captures its “subjective

perception” about the aggregate state. The firm uses its private signal to make inferences

on productivity and money supply shocks that affect marginal costs and aggregate demand.

Furthermore, the signal is used to infer the beliefs, and therefore the pricing decisions, of

other firms. An entire hierarchy of beliefs about the price level is formed and will be the

topic of a later section. Thus, price-setters face both fundamental and strategic uncertainty.

Overconfidence enters the inference problem by making agents put too much weight on their

own subjective perception of the aggregate shock. Moreover, we allow the fraction of un-

informed firms to be endogenous : each firm j can pay a real fixed cost cj and obtain full

information about the shock θ and the pricing decisions of the rest of the firms.

The timing is as follows. The firm receives a private signal. Given the signal, the firm

decides to obtain or not information. If the firm becomes “informed”, it sets price P †(j). If

the firm decides to stay “uninformed,” it sets price P (j). We can characterize this problem

by working backwards.

Informed firm. The optimal price when the firm has complete information is just the

typical markup over nominal marginal cost, which can be written as in (8), P †(j) = P 1−ηP ∗η.

The price level P is obviously not the same as in an environment with full information.

Uninformed firm. If the firm does not obtain information, it sets a price that maximize

its subjective expectation of real profits. Let Ej denote the subjective expectation operator

of firm j, given its private information set. P (j) maximizes Ejπ(P (j), P, θ) which implies

that

P (j) =Ej{P †(j)Z}Ej{Z}

, (9)

where Z is defined as Z ≡ MP ε−2 and P †(j) is given by (8). According to (9), when

there is incomplete information, a generic firm j sets its price as a subjective expectation

(appropriately weighted) of nominal marginal cost, i.e. as an average of the price that would

be set under complete information.

6

Obtaining information. A generic firm j chooses to acquire complete information when

the expected increase in profits in doing this is higher than the cost cj

Ej{π(P †(j), P, θ)− π(P (j), P, θ)} ≥ cj. (10)

Having observed the realization of its own signal, a firm j evaluates the left-hand side

(LHS) of (10). The firm acquires information and sets price P †(j), if (10) holds, otherwise it

chooses P (j).

3.3 Approximation to the incomplete-information model

We have not specified yet the signal structure and the filtering problem that is hiding behind

the expectation operator Ej. Before we do that, we will simplify the problem with an

approximation around a deterministic steady state where θ = θ = (A, M).

Proposition 1. As second-order approximation to the LHS of (10) leads to a criterion of

the form

varj{p†(j)} ≥ cj, (11)

where varj{·} denotes the variance operator conditional on the subjective information set,

while p†(j) is the log of the price that firm j would set with complete information and cj is a

reparametrization of the cost cj.

Proof. See Appendix.

The decision of acquiring or not information depends on whether the subjective variance

of the price that a firm sets under complete information is higher than the cost cj. An

important implication of proposition 1 is that (11) can be evaluated using just a log-linear

approximation to the equilibrium conditions. In this log-linear approximation, equation (9)

implies that the log of the price under incomplete information is the expected value of the

log of the price under complete information

p(j) = Ejp†(j), (12)

where lower-case letters denote log of the respective variables.4 Moreover p†(j) is independent

of j and from (8) it can been seen that it satisfies an exact log-linear relationship given by

p† = (1− η)p+ ηp∗, (13)

4In the steady-state all firms set the same constant price.

7

where p∗ is the log of the full-information equilibrium price level and p is given by

p =

∫ 1

0

p(i)di

as a result of a first-order approximation of (3).

Let us denote with µ the fraction of firms that in equilibrium decide to keep the subjective

information set and assuming without loss of generality that agents j ∈ [0, µ] are the ones

who remain uninformed, we can write the above equation as

p = µp+ (1− µ)p†, (14)

where we have defined with p the average price of the subjectively-informed firms as

p ≡ 1

µ

∫ µ

0

p(i)di. (15)

We can then plug (14) into (13) to obtain

p† = δp∗ + (1− δ)p, (16)

where

δ ≡ η

η + (1− η)µ.

The weight on the full information price is equal to the degree of complementarity, δ = η,

if every firm is uninformed (µ = 1) as in Woodford (2002). The weight δ is a decreasing

function of µ and η ≤ δ ≤ 1 in case of strategic complements (η < 1) and δ an increasing

function of µ with 1 ≤ δ ≤ η in case of strategic substitutes (η > 1).

The set of equations (12), one for each firm that remains uninformed, together with (15)

and (16) determine the equilibrium prices of informed and uninformed firms in a first-order

approximation to the equilibrium conditions.

3.4 Signal extraction and overconfidence

We describe now the information structure in detail. We note that uncertainty about the

aggregate shock θ has collapsed to uncertainty about the full-information price p∗, which

is a linear combination of the logarithms of the technology and the money supply shock,

p∗ = const. + lnM − 1η

lnA. Without loss of generality for our purpose of exploring the

effects of overconfidence on pricing, we are going to assume that firms receive signals about

8

this particular linear combination p∗. Let p∗ be a random variable of the form

p∗ = p∗ + u,

where p∗ is a constant and u is a Gaussian white-noise process with variance σ2u. This

distribution is common knowledge and corresponds to the objective probability distribution

of p∗.

Overconfidence. Each firm receives a private signal sj that is linearly related to p∗ as

sj = p∗ + ξj,

where ξj is an idiosyncratic Gaussian noise with mean zero. Let σ2ξ and σ2

ξ denote the perceived

and true variance of ξj. We introduce overconfidence as in the influential contributions of

Daniel et al. (1998) and Daniel et al. (2001) in the behavioral finance literature: we assume

that σ2ξ < σ2

ξ . So price-setters overestimate their ability to infer the hidden state from their

private signals. This corresponds to the behavioral bias of overprecision.5 Moreover, ξj is

statistically independent of u as well as of ξi for each i 6= j. All this information is common

knowledge.

Given this information structure, each firm can form its own expectation of the full-

information price p∗ as in a standard signal-extraction problem

Ejp∗ = (1− rj)p∗ + rjsj, (17)

where the weight rj is defined as

rj = r ≡ 1

1 + λwith λ =

σ2ξ

σ2u

.

Since λ is common and common knowledge across the different firms, then rj is indepen-

dent of j and equal to a common r. In particular, λ represents the noise-to-fundamental

variance ratio and can be interpreted as an index of confidence in how a firm’s private signal

is a good representation of the full-information price. Lower values of λ implies a higher

weight to the signal when firms form expectations of the full-information price and then a

high degree of confidence on the subjective information set. We can define a ‘true’ degree

of confidence λ as λ ≡ σ2ξ

σ2u

with a respective weight in the filtering problem, r = 1/(1 + λ).

5See the survey of Moore and Healy (2008) for a succinct taxonomy of overconfidence studies in terms ofoverestimation, overplacement and overprecision.

9

Under overconfidence, we have λ < λ and therefore r > r, so there is an excessive weight on

the private signal.

3.5 Higher-order beliefs and optimal pricing

Each firm forms its own expectation of the signals of others as

Ejsi = Ejp∗ = (1− r)p∗ + rsj

which is then a first-order expectation belief.6 Furthermore, each firm can form its subjective

expectation of others’ first-order expectation belief as

EjEisk = Ej[(1− r)p∗ + rsi] = (1− r)p∗ + rEjp∗

and so on.

To solve for the equilibrium prices of informed and uninformed agents and for the equi-

librium fraction of firms that acquire or not information, we first guess that µ is known to

each firm j. We then verify that this is indeed the case. Given this guess, we can substitute

(16) into (12) to get

p(j) = δEjp∗ + (1− δ)Ej p (18)

which can be averaged across all uninformed price setters to obtain

p = δEp∗ + (1− δ)Ep,

where we have defined the operator E(·) ≡ 1µ

∫ µ0E(·)di. This operator represents the average

expectation among the uninformed firms. We can iterate the above expression to obtain

p = δ

∞∑k=0

(1− δ)kE(k+1)p∗, (19)

where the (k+1)−order average expectation operator is defined as E(k+1)(·) ≡ E(E(k)(·)) for

each k ≥ 1.7 It follows that the average price of uninformed firms is a linear combination of

their higher-order average expectations of the full-information price. We can plug (19) into

6See Allen et al. (2006), Amato and Shin (2003) and Amato and Shin (2006) for examples of problemswith iterated expectations.

7Under the restriction that η(2µ− 1) < 2µ , δ is such that |1− δ| < 1.

10

(18) to obtain

p(j) = δEjp∗ + δ(1− δ)∞∑k=0

(1− δ)kEjE(k+1)p∗.

The price set by an uninformed firm depends on its own expectation of the full-information

price and its own expectation of the average expectation (of uninformed firms) of the full-

information prices as well as on all higher-order expectations. Since (17) holds for all unin-

formed firms, we can obtain that

Ep∗ = (1− r)p∗ + rp∗, (20)

where we have implicitly assumed that the law of large number holds on a positive measure8

1

µ

∫ µ

0

ξidi = 0. (21)

It follows that the firm i’s expectation of the average expectation of the full-information

price is given by

EiEp∗ = (1− r)p∗ + rEip∗

= (1− r)p∗ + r(1− r)p+ r2si

from which it follows that the second-order average estimate is given

E(2)p∗ = (1− r)(1 + r)p∗ + r2p∗.

By re-iterating the above arguments, we get that the k-fold average expectation of the full-

information price is

Ekp∗ = (1− rk)p∗ + rkp∗. (22)

We can substitute (22) into (19) to finally obtain

p =1− r

1− (1− δ)rp∗ +

rδ

1− (1− δ)rp∗

=η(1− r) + (1− η)(1− r)µ

η + (1− η)(1− r)µp∗ +

rη

η + (1− η)(1− r)µp∗. (23)

8See Uhlig (1996) for the conditions under which this holds.

11

Substituting (23) into (16) we obtain that the price set by any informed firm is

p† =(1− r)(1− η)µ

η + (1− η)(1− r)µp∗ +

η

η + (1− η)(1− r)µp∗. (24)

3.6 Information acquisition

Given (24), it follows that a generic firm j decides to acquire information if the following

inequality holds [η

η + (1− η)(1− r)µ

]2

varj{p∗} ≥ cj,

where varj{p∗} is the variance of the full-information price level conditional on the subjective

information set of firm j. This is given by

varj{p∗} = σ2u(1− rj).

We can then write the above inequality as[η

η + (1− η)(1− r)µ

]2

σ2u(1− rj) ≥ cj, (25)

where we have kept the distinction –since it matters for the discussion that follows- between

the own degree of confidence rj and the others’ degree of confidence r– although we have

assumed that they are the same.9

According to (25), several parameters of the model drive the incentives for firm j to

acquire information. The higher is the prior on the variance of the full-information price,

σ2u, the higher are the incentives to acquire information. Obviously, the lower the cost cj,

the higher those incentives. In the case of strategic complementarity in the pricing decision,

0 < η < 1, the higher is the fraction of firms that are acquiring information (i.e. the lower

the µ) the higher are the incentives for the individual firm to acquire information. This result

is of the same nature as the one found by Ball and Romer (1989) in a similar model but with

only imperfect information, in which firms’s decisions are on whether to change or not prices.

Each firm’s decision is also influenced by the degree of confidence in the informativeness

of the signal. If rj is high (λj is low), then the firm will not have incentives to acquire finer

information. A high degree of confidence implies that firms are going to be stuck with their

perceptions of the world when setting their prices.

Interestingly, if the confidence of others increases (λ decreases and r increases) then the

9Indeed, (24) could have been derived even if we didn’t have the same ri by defining r ≡(∫ µ

0ridi

)/µ and

by assuming a law of large numbers to hold for∫ µ0ξiridi = 0.

12

price under complete information has higher subjective variance since the average price of

uninformed firms is getting close to the full information price, as shown in (20). Then, each

individual firm has higher incentives to acquire information and imitate other firms –when

pricing decisions are strategic complements.

We move to characterize the equilibrium value of µ, under the assumption rj = r for each

j. We define

c∗ ≡

[η(1− r) 1

2

η + (1− η)(1− r)µ

]2

σ2u

and note that (25) implies that all firms with cj less than c∗ acquire information. Let the

cumulative distribution function of costs be F with respective density f on support [c , c].

The fraction µ of uninformed firms satisfies

1− F (c∗) = µ. (26)

This solution confirms our initial guess that µ is a function of known parameters and then

known to each firm j. The properties of F (·) determine the existence and the characteristics

of the equilibrium. Indeed, when cj = c for each j, multiple equilibria are possible for the

same reasons as they occur in the imperfect-information model of Ball and Romer (1989).

For other F (·) multiple equilibria might disappear. Since this is not the focus of this work,

we assume that F (·) and f(·) are such that there exists a unique equilibrium. We get the

following proposition.

Proposition 2. (“Overconfidence and uninformed firms”)

Assume that we are at a stable equilibrium where 1 + f(c∗)∂c∗

∂µ> 0.

If (1− η)(1− r)µ < η thendµ

dr> 0. (27)

Thus, the fraction of uninformed firms increases when the weight on the private signal

increases.

Proof. See Appendix.

The proposition implies that overconfidence can lead to a larger mass of uninformed firms

relative to a rational signal-extraction benchmark.

Discussion. Proposition 2 is not a trivial consequence of the overconfidence of the agents.

It involves the effects of firm’s own confidence and the opposing effects of the confidence

13

of other firms through the mechanism of higher-order expectations. An increase in one’s

confidence decreases the incentives to acquire information. But an increase in the confidence

of others makes the price of informed firms more volatile, since there is larger reliance on

private signals in equilibrium, amplifying the effect of higher-order beliefs. To see that,

consider the extreme case where private signals are completely uninformative, r = 0. In that

case, the entire mechanism of higher-order beliefs is mute: uninformed firms set a price equal

to their prior, p = p∗, and the price of informed firms becomes p† = (1− δ)p∗ + δp∗. So the

weight on the full information price p∗ reaches its minimum, leading to small volatility and

reduced incentives to acquire information.

The inequality condition in (27) requires that the effect of the firm’s own confidence,

which leads to a larger equilibrium fraction µ, is stronger than the higher-order beliefs effect,

which reduces µ. As expected, the condition always holds in the case of strategic substitutes,

η > 1. In the case of strategic complements, the condition holds if we effectively limit the

effect of higher-order beliefs. This would happen in an equilibrium where µ is small (so the

complementarities are not strong enough), µ < η(1−η)(1−r) . From another angle, the higher-

order beliefs effect would be contained if we bounded η away from a lower bound, by writing

the condition as η > (1−r)µ1+(1−r)µ . The lower bound for η is always smaller than 1/2 and decreases

to zero when r increases to unity. The conclusion is that if complementarities are not too

“large,” overconfidence increases the equilibrium fraction of uninformed firms.

4 Price implications of incomplete information and over-

confidence

In the previous section, we have shown how to determine the fraction of firms that in equi-

librium decide to remain uninformed. A high degree of confidence under standard regularity

conditions implies a higher fraction of uninformed firms. In this section we study the price

implications of the model and in particular the relation between excess volatility of prices

and overconfidence.

A first important implication is that the model displays two levels of heterogeneity: at

a first stage, there are differences in prices between informed and uninformed firms. At

a second stage, within uninformed firms, prices are related to the realization of subjective

signals. We can rewrite equation (24) and show that the price of the informed firms is

p† = p∗ + (1 + λ)ku, (28)

14

where

k =rη

η + (1− η)(1− r)µ.

The prices of the informed firms react only to the fundamental shocks of the model. On the

opposite the uninformed firms set their prices as a subjective expectation of p†, based on their

signals which include also non-fundamental noises. We obtain that a generic uninformed firm

j sets

p(j) = p∗ + ku+ kξj, (29)

while the average price of uninformed firms is given by

p = p∗ + ku.

We first discuss how prices react to fundamental shocks. Following (28) prices of informed

firms react less than proportionally to fundamental shocks when pricing decisions are strategic

complements, since (1 + λ)k < 1, but more than proportionally in the strategic-substitute

case. As shown in (29), the response of uninformed firms is always smaller than that of

informed firms, since λ > 0. The dicrepancy is coming from the fact that the informed firms

do not have to filter the hidden shock.

However, prices of uninformed firms react also to non-fundamental shocks, ξj, in the same

proportion as they do to fundamental shocks.

Overconfidence can affect the volatility of prices. Using equation (29), we obtain that the

“true” variance of prices for a generic uninformed agent j is

var{p(j)} = (1 + λ)k2σ2u.

Equation (28) implies that the variance of the prices of informed firms is given by

var{p†} = (1 + λ)2k2σ2u.

It follows that the ratio of the volatilities of prices of uninformed and informed firms is

given by

var{p(j)}var{p†}

=

[(1 + λ)

12

(1 + λ)

]2

.

When the signal-extraction problem is rational (i.e. λ = λ), prices of uninformed firms are

always less volatile than informed firms. With overconfident firms, it is instead possible for

the reverse to happen. It is sufficient that (1 + λ) < (1 + λ)12 , which requires that the true

volatility of the idiosyncratic noise σ2ξ is large enough relative to the perceived σ2

ξ .

15

A second important implication of overconfidence is that it is even possible to have excess

volatility of the price of an individual uninformed firm with respect to the full-information

(fundamental) price. Indeed, we obtain that

var{p(j)}var{p∗}

=

[ζ(1 + λ)

12

(1 + λ)

]2

where ζ is a positive parameter given by ζ = k/r such that ζ < 1 (ζ > 1) when pricing

decisions are strategic complements (substitutes). To have excess volatility of the prices of

uninformed firms with respect to fundamentals, it is required that (1 + λ) < ζ(1 +λ)12 which

is then a more (less) stringent condition than before when pricing decisions are strategic

complements (substitutes).10

Overconfidence has two important roles in this model. On one side, it implies that a

higher fraction of firms is going to decide optimally not to acquire information and just pay

attention to their own private signals. On the other side, the prices of uninformed firms can

be more volatile than fundamental disturbances and this volatility is driven by the noise in

the perception of the full-information price. In the dynamic extension of the above model,

the fact that overconfident price setters are less prone to acquire information implies that

there can be a high proportion of this kind of subjectively-driven price setters. Woodford

(2002) has shown that higher-order expectations matter for determining persistent effects of

output and prices following exactly those shocks agents are subjectively informed about. In

our model, the existence of subjectively-informed firms with overconfident beliefs can produce

an excess volatility of prices with respect to fundamentals.

5 Infinite-horizon model

In this section, we consider an extension of the previous model to an infinite horizon. We

assume that each firm does not know the realization of the sequence {θt}∞t=t0 . However,

each firm has a prior distribution on the sequence {θt}∞t=t0 that coincides with the correct

distribution and which is common knowledge. In each period and contingency, each firm can

observe a private signal sjt . In particular the sequence of signals {sjt}∞t=t0 , one for each j, is

related to the sequence {θt}∞t=t0 through a likelihood function which is known and common

knowledge but, as before, does not necessarily coincide with the correct likelihood function.

As in the previous model, incomplete information is modelled by assuming that each firm

10Note that with no overconfidence (λ = λ) the ratio is always smaller that unity even in the case of

strategic substitutes (ζ > 1). This is clear if we note that k = δ/(δ + λ) and that the ratio is less than unitywhen Q (δ) = λδ2 − 2λδ − λ2 < 0, which holds for the permissible δ, i.e. such that |1− δ| < 1.

16

knows only its own realization of the signals and not those of the others, as well as not the

price index and the individual prices. Each firm has the option to acquire information by

paying a cost cj. Once the cost is paid, the firm remains in the ‘informed’ state forever.

We assume that firms choose prices to maximize the expected discounted value of profits

given by

Ejt0

∞∑t=t0

βt−t0π(Pt(j), Pt, θt),

where β is such that 0 < β < 1.11 Ejt0 is the appropriate expectation operator conditional on

the information at time t0. Prices are set freely in each period. An ‘informed’ firm sets its

price as

P †t (j) = P 1−ηt P ∗ηt

for each period t after having paid the information cost. An ‘uninformed’ firm instead sets

its price as

Pt(j) =Ejt {P

†t (j)Zt}

Ejt {Zt}

,

where Zt has the same definition as before and the expectation operator is conditional on

the information set of firm j at t.

To characterize the decision for a generic firm j to acquire or not information, we guess

an equilibrium and then verify that prices and information decisions are consistent with

that equilibrium. The analysis is simplified by noting that the fraction of firms that remain

uninformed each period cannot increase over time, i.e. {µt}+∞t=t0 is a non-increasing sequence.

Of the many equilibria that can exist, we are interested in ones in which µt = µ for each

t ≥ t0. In particular, in these stationary equilibria, whichever firm decides to be informed

does it in the first period. For this to be optimal, the strategy of getting information in the

first period should give higher expected discounted profits than the strategy of waiting until

a generic time T , given the equilibrium strategies of all other firms. In particular at time t0

the expected profits to acquire immediately information and pay the cost should be higher

than the strategy of remaining with the subjective information until a generic period T and

pay the cost in that period. For a generic firm j to become informed at time t0, the following

inequality should hold for each T > t0

Ejt0

T−1∑t=t0

βt−t0π(Pt(j), Pt, θt)− βT cj ≤ Ejt0

T−1∑t=t0

βt−t0π(P †t (j), Pt, θt)− cj,

11We can generalize the analysis that follows by assuming a stochastic discount factor to evaluate realprofits across contingencies and time.

17

which can be rewritten

Ejt0

T−1∑t=t0

βt−t0{Ejt [π(P †t (j), Pt, θt)− π(Pt(j), Pt, θt)]

}≥ cj(1− βT ). (30)

We take a second-order approximation of the above problem around a stationary point

with unitary relative prices to obtain

Ejt0

T−1∑t=t0

βt−t0varjt{p†t} ≥ cj(1− βT ).

We guess, and verify later, that in the equilibrium varjt{p†t} is a constant that does not depend

on j and is also independent of t in a stationary filtering problem. The above condition then

simplifies to

varjt{p†t} ≥ cj(1− β), (31)

which is also independent of T.

We verify now that varjt{p†t} is constant and independent of j, and that µ is also a

constant and known within the information set of each type of firm at time t0. As before,

we just need to characterize the equilibrium values of prices in a log-linear approximation to

the equilibrium. It is still true that the set of equations (12), one for each firm that remains

uninformed, together with (15) and (16) determine the equilibrium prices of informed and

uninformed firms in a first-order approximation to the equilibrium conditions. We continue

to assume that each firm receives a private signal sjt that is related linearly to p∗t as

sjt = p∗t + ξjt , (32)

where ξjt is an idiosyncratic Gaussian noise with mean zero, perceived variance σ2ξ and true

variance σ2ξ for each j, with σ2

ξ < σ2ξ . We assume that ξjt , for each j, is statistically indepen-

dent of the sequence {p∗t} as well as of the sequence {ξit} for each i 6= j. All this information

is common knowledge, but the realization of the private signal sjt is private information. We

allow now {p∗t} to be a first-order autoregressive stochastic process of the form

p∗t = p∗ + ρp∗t−1 + ut (33)

with |ρ| ≤ 1 where ut is Gaussian noise with mean zero and variance σ2u. The assumption

of persistence of the unobservable shock can in principle be a source of complication in the

solution of the model, for an infinite dimensional state might be necessary to keep track of

18

the higher-order beliefs of other firms. Woodford (2002) has shown that the dimension of the

hidden-state space is finite in the same model as the one presented here but with all firms

assumed to be uninformed and without any endogenous decision of acquiring information.

Proposition 3. (“Optimal prices in the dynamic model”)

• The general price index evolves according to

pt = p∗ + ρ(1− k)pt−1 + ρkp∗t−1 + [δ (1− µ) (1− k) + k]ut,

where 12

k ≡ 1

2ρ2

ρ2 − 1− δ

λ+

√[(1− ρ2) +

δ

λ

]2

+ 4ρ2δ

λ

.

• The price of informed firms follows

p†t = p∗ + ρ(1− k)p†t−1 + ρkp∗t−1 + [δ(1− k) + k]ut. (34)

• The price of uninformed firms follows

pt(j) = p∗ + ρ(1− k)pt−1(j) + ρkp∗t−1 + k(ut + ξjt ). (35)

• The contemporaneous variance of p†t in a stationary solution is given by

varjt{p†t} =

1 + λ[1− ρ2(1− k)]2

[1− ρ2(1− k)2]λk2σ2

u. (36)

Proof. See the Appendix for the details of the calculations. Our approach is a variant of

Woodford (2002), properly modified to account for the fact that there is a fraction of firms

that is uninformed and acquires endogenously information.

Note that the static model is nested under the assumption that ρ = 0. We can now

evaluate (31). It follows that equilibrium fraction of uninformed firms is implicitly defined

12The parameter k in the dynamic model is a different function of other (exogenous) parameters than the

k in the static model and represents a linear combination of the vector of Kalman gains. We use the samenotation, since when ρ = 0 the two expressions coincide.

19

by the same condition as (26) where now instead c∗ is given by

c∗ ≡ 1 + λ[1− ρ2(1− k)]2

[1− ρ2(1− k)2]

λk2σ2u

(1− β). (37)

The main qualitative results of the static model hold in this extension with some qualifica-

tions. Indeed it is still the case that overconfidence is needed for the volatility of prices of

uninformed to be higher than that of informed. The ratio of the unconditional variances

between informed and uninformed firms is higher than the unitary value when the following

criterion holds:

Proposition 4. (“Excess price volatility”)

var{pt(i)}var{p†t}

> 1

if and only if

λ > 2λ+ λ2[1− ρ2(1− k)2]. (38)

Proof. See Appendix.

Note that the criterion nests the static case result for ρ = 0. In this dynamic model, it does

not only matter the difference between the ‘true’ and the subjective degree of confidence,

but also other parameters. Indeed since k < 1, the discrepancy between λ and λ that is

needed in order to have excessive volatility of the uninformed prices is smaller than in the

static case. The reason is that the persistence of the shock process makes past estimates

useful to forecast the future evolution of the state. But this leads to a larger reliance on their

subjective perception (signals) and therefore, comparatively to the static case, agents are

driven more by their perceptions. As a consequence, the amount of overconfidence needed

to have excess volatility is less. This is also the case if the mass of uninformed agents (µ)

increases since k becomes smaller and if the degree of strategic complementarity increases,

i.e. η becomes smaller.

6 An application

In this section we explore the empirical implications of our model for individual price volatility

and aggregate inflation persistence. There are two empirical facts of interest. First, Klenow

and Kryvtsov (2008) have shown that the average absolute size of price changes for individual

goods composing the CPI index amounts to 8.5% on a monthly basis. Second, VAR analyses

20

on the response of output to monetary disturbances have shown a hump-shaped behavior

with a peak that ranges from 4 till 6 quarters like in Christiano et al. (2005).

There are many possible sources for price volatility which we have completely abstracted

from in our analysis, like idioyncratic productivity or demand shocks. Our objective here

is to show that the mechanism presented in this paper can in some ways complement other

explanations. To this end, we ask to what extent, under a reasonable calibration of the

model, degrees of overconfidence found in experimental studies can explain the variability of

prices found in the data. With reasonable calibration we mean that our model should be also

consistent with some other moments of the data reported in Klenow and Kryvtsov (2008)

and at the same time require information costs which are in line with other empirical studies.

Moreover, we want to show that our explanation can be consistent with a model that allows

for a persistent response of output to a nominal spending disturbance.13

To start the exercise, we note that Bils and Klenow (2004) found that prices are sticky

with a median duration of 4.3 months. But in our model firms change their price in each

period, even if they have incomplete information. For that reason, we decide to measure the

time period of our model in quarters, during which we might reasonably assume that all the

firms had the time to adjust their prices.

The parameters of the model are (ρ, p∗, σ2u, λ,

λλ, η, ε, F ), where F indicates the parameters

that characterize the distribution of the costs. We calibrate ρ = 1 in a way that our model

can be consistent with a unit root in the general price index, which in general is not rejected

by the data.14 Under this assumption equilibrium output in deviation from the steady state

(i.e. yt ≡ lnYt/Y ) follows the process

yt = (1− k)yt−1 + (1− c)ut,

where c = (δµ + 1 − δ)k + δ(1 − µ) and δ = η/(η + (1 − η)µ). So k is a parameter that

summarizes the output persistence to a monetary shock like in Woodford (2002). From this

process we can deduce that the half-life of the response of output to a nominal spending

shock is given by τ = − ln 2/ ln(1 − k). Since we want our model to display a persistent

response of output and at the same time be consistent with all the other facts, we assume

that τ is equal to 4 quarters as it is suggested by the VAR literature and which is a reasonable

value according to the discussion of Woodford (2003, ch. 3). This assumption implies that

13For calibration purposes we are going to treat the shock in the full infomation price as a nominal spendingshock.

14We have experimented with data on the non-shelter CPI for the sample period of KK (1988:1-2003:4)and found that a null of a unit root in the price level cannot be rejected.

21

k = 0.159. Under the assumption of ρ = 1, the process of the inflation rate is given by15

πt = kp∗ + (1− k)πt−1 + cut − (c− k)ut−1. (39)

In their sample, KK find an average monthly percentage price change (excluding sales) among

firms that change prices of 1.11 % with a standard deviation of 1.14%. This corresponds to a

calibration of p∗ = 1.11 for the mean price change among all firms in our model. Furthermore,

their data on the variance of the inflation provide an additional moment condition to match

for the variance of the model inflation rate, σ2π, which according to (39) is just a function of

the form

σ2π = f(µ, k, η, σ2

u) (40)

pinning down a relation between η, σ2u and µ for given k, σ2

π.

Studies on the costs of price adjustment like Zbaracki et al. (2004) have shown that

managerial and customer costs of price adjustment constitute a large fraction of firms profits

even if actual menu costs are quite small. In fact, managerial costs (which refer to information

gathering, decision-making and communicating-to-sales-team costs) are 4.61% of the profits,

whereas customer costs (communication and negotiation costs) can reach even 15.01% of

profits. The concept of managerial costs is the closest to the notion of information-processing

costs in our model and therefore we use the figure of 4.61% of profits as the most relevant

as an average measure of cost. More specifically, we assume a uniform distribution of costs

per period as a fraction of steady state profits with a minimum cost of zero and a mean cost

of 4.61%.16 Furthermore, we assume an elasticity of substitution between the differentiated

products ε = 6, which corresponds to a markup under full information of 20%. Given these

assumptions, and noting that λ = δ(1 − k)/k2 equations (26) and (37) imply the following

relation

g(µ, k, η, σ2u) = 0 (41)

which is then another implicit relation between η, σ2u and µ for given k.17

The parameter η is critical for determining whether pricing decisions are strategic com-

plements or substitutes and plays a crucial role in determining the persistence of the response

of output to a monetary shock, as discussed in Woodford (2003). Indeed, Woodford (2003,

ch. 3) has shown that when a sufficient degree of strategic complementarity is assumed, i.e. a

low value of η around 0.15, then a sticky-price model can account for a prolonged response of

15Note that since the idiosyncratic noise washes out in the aggregate, the amount of overconfidence doesn’taffect the aggregate dynamics of inflation and the relevant mean and standard deviation.

16Note that the maximum cost in this parametrization (9.22%) is still smaller than the measure of customercosts of Zbaracki et al. (2004).

17See details in the Appendix.

22

output. On the opposite case, Chari et al. (2000) assume a value of η equal to 2.25, leading

to pricing decisions that are strategic complements and then have argued that a sticky-price

model is not able to generate enough persistence. In this work, we take an agnostic view

on η and experiment how our results may differ for a range of values for this parameter.

In particular, we report results for a range of η that goes from the low number assumed

in Woodford (2003, ch. 3) to the high number used in Chari et al. (2000). Having fixed η,

then equations (40) and (41) determine µ and σ2u. We can determine δ and, bearing in mind

that λ = δ(1 − k)/k2, we can also determine the degree of confidence compatible with our

calibration strategy.

Table 1 presents the results of the calibration and in particular how the parameters and

the equilibrium fraction µ are influenced by the chosen value of η. We observe that lower

values of η require higher variance of the fundamental shock σ2u to match the variance of

price changes in the data and at the same time have the model consistent with a half-life of 4

quarters and reasonable information costs. However, the degree of confidence on the signals,

which is λ, that measures how good the signals are as a proxy of the fundamental shock, is

low for low values of η. This perhaps indicates that the model is more reasonable when η

is in this range and a sufficient degree of price complementarity is assumed. Using η = 0.15

we obtain that 83% of firms maintain a subjective information set when setting their prices.

This value decreases as η increases and reaches 57% when η = 2.25.

Table 1: Calibration and Equilibrium Fraction µ

η =0.15 η =0.5 η =0.9 η =1.1 η =1.5 η =1.9 η =2.25

p∗ 1.11 1.11 1.11 1.11 1.11 1.11 1.11

σu 4.234 2.97 2.310 2.131 1.903 1.764 1.678

λ 5.81 20.6 31.20 35.05 41.07 45.53 48.62

µ 0.831 0.611 0.580 0.577 0.573 0.571 0.570

We then proceed to simulate the model in order to characterize the pricing behavior of

20000 firms over the sample 1988:1-2003:4 as in KK. Over this sample, we compute the

same statistics that they report and in particular we focus on the average over the sample

and across firms of the absolute of the changes in the individual prices for two subsequent

observations. This statistic corresponds to the statistic |dp| for regular prices of all items in

their Table 1. By construction, our simulations are in line with the statistics that they report

23

on the mean and variance of the price changes.18

Our results on the variable |dp| depend on the degree of overconfidence assumed. We

have defined overconfidence as the ratio λ/λ which is equivalent to σ2ξ/σ

2ξ , the ratio of the

true variance of the noise with respect to the perceived variance. We choose as an index of

overconfidence the parameter γ = σξ/σξ which then gives a measure on how much the true

standard deviation of the noise exceeds the believed one. We repeat our simulations for the

chosen values of η by letting γ varies from 1 to 8. In particular γ = 1 corresponds to the

rational signal extraction problem. The results are presented in Table 2.

Table 2: Average absolute value of price changes (%) and overconfidence

η =0.15 η =0.5 η =0.9 η =1.1 η =1.5 η =1.9 η =2.25

γ = 1 1.83 2.012 1.99 1.97 1.95 1.94 1.93

γ = 2 2.75 3.02 2.89 2.86 2.8 2.76 2.73

γ = 3 3.80 4.05 3.86 3.79 3.69 3.63 3.59

γ = 4 4.88 5.13 4.83 4.73 4.6 4.51 4.46

γ = 5 5.98 6.21 5.81 5.68 5.52 5.40 5.33

γ = 6 7.07 7.28 6.79 6.65 6.43 6.30 6.21

γ = 7 8.18 8.37 7.77 7.60 7.36 7.19 7.08

γ = 8 9.30 9.45 8.76 8.56 8.28 8.08 7.96

Table 2 shows for each pair γ and η the average absolute value of price changes (|dp|) implied

by this model. The reference value is the 8.5% of KK. In a rational signal extraction problem

(γ = 1) we obtain a value close to 2%. But to match their reported value we need a

degree of overconfidence close to 7 or 8, which can be considered as a large number. Indeed,

experimental studies, like Soll and Klayman (2004), have shown that on a series of questions

where individuals are asked to form an 80% confidence interval the actual hit rate is around

40%, which can be translated in γ being approximately equal to 2.5.19 In more complicated

tasks, as forecasting the level of the exchange rate with a confidence interval of 90% (see

Oberlechner and Osler (2009) and Oberlechner and Osler (2012)), the hit rate ranges from

5% to 70% with an average of 40%, rationalizing values of γ higher than 3. In general values

from 2 to 4 can be considered as reasonable.

18We have repeated the described simulation 1000 times to smooth out any influence of the small sampleon the results and we average the statistics across these repeated simulations.

19This value can be obtained by rough computation on confidence intervals for normal distributions.

24

If we set our target less ambitiously and ask what is the implied |dp| for a degree of

overconfidence close to experimental evidence, we find that this can range from 3% to 5%,

doubling the value under a fully rational model and being close to explain more than the 50−60% of the value found in the data.20 We think that these results suggest that our proposed

mechanism can have some value in explaining price volatility –although we acknowledge that

there can be other important mechanisms from which we have abstracted in this analysis.

7 Concluding remarks

In this paper, we study the behavior of individual and aggregate prices in an economy with

monopolistic-competitive firms that is driven by aggregate shocks observed with noise. Each

firm receives a private signal about the hidden state. We assume that firms are overconfi-

dent in their signals, that is, their overestimate the precision of their own perception of the

aggregate state.

This model can rationalize a persistent response of output to the aggregate hidden state

and be consistent at the same time with excess volatility of individual prices, providing

therefore a unified “noise” interpretation of aggregate and idiosyncratic prices. We see our

approach as complementary to setups where idiosyncratic fundamental shocks are the main

driver of price volatility.

More generally, we believe that behavioral biases, especially in the processing and analy-

sis of information, are not easy to dismiss. The way forward is to conduct careful studies of

actual decision makers in real market conditions. For example, Bachmann and Elstner (2015)

use confidential German manufacturing survey data and document substantial instances of

systematic positive and negative expectation errors. Further studies that carefully docu-

ment second-moment biases are necessary in order to clarify the quantitative importance of

overconfidence.

20 As Table 1 shows, this result does not depend on the value of η assumed.

25

A Static model

A.1 Proof of proposition 1

Derivation of condition (11) from (10). We first note that by using condition (7) wecan rewrite marginal costs in terms of the complete information price and by using alsocondition (9) we can express the expected profits as functions of P †(j) and P (j) and thevariable Z ≡MP ε−2 as follows

Ej{π(P †(j), P, θ)} =1

εEj{

(P †(j)

)1−εZ}, (A.1)

Ej{π(P (j), P, θ)} =1

εP 1−ε(j)EjZ. (A.2)

We take a second order approximation of expected profits around a deterministic steadystate where θ = θ and as a result P = P † = P ≡ P . Let lowercase variables denotelog-deviations from the steady state and let ‖p‖ and ‖z‖ denote a bound on the size offluctuations for the price of each differentiated good and for the variable Z respectively. Wecan obtain by approximating equation (A.1) that

Ej{π(P †(j), P, θ)

}=

1

εP 1−εZ

[1 + (1− ε)Ejp† +

1

2(1− ε)2Ej

(p†)2

+

+Ejz +1

2Ejz2 + (1− ε)Ejp†z

]+O(‖p, z‖3). (A.3)

Similarly by approximating equation (A.2) we obtain that

Ej{π(P (j), P, θ)} =1

εP 1−εZ

[1 + (1− ε) p (j) +

1

2(1− ε)2 p (j)2

+Ejz +1

2Ejz2 + (1− ε) p (j)Ejz

]+O(‖p, z‖3). (A.4)

Note that P 1−εZ = MP

= Y and let

W (j) ≡ π(P †(j), P, θ)− π(P (j), P, θ)

denote the difference in profits. Then using (A.3) and (A.4) we have

Ej{W (j)} =ε− 1

εY

{p (j)− Ejp† +

1

2(ε− 1)

[Ej(p†)2 − (p (j))2

]−Ej[

(p† − p (j)

)z]}

+O(‖p, z‖3). (A.5)

Note that the price of the uninformed agents (1) in a first order approximation is

p (j) = Ejp† +O(‖p, z‖2). (A.6)

26

Furthermore, if we take a second-order approximation of it we obtain

p (j)− Ejp† =1

2V arj

(p†)

+ Ej[p† − Ejp†

]z +O(‖p, z‖3), (A.7)

where V arj(p†) ≡ Ej(p†)2 − (Ejp†)2. Using (A.6) and (A.7) into (A.5) we observe that theterms involving z cancel out and that Ej(p†2 − p(j))2 = V arj(p†), so ignoring third orderterms we obtain

Ej{W (j)} =1

2

ε− 1

εY {V arj(p†) + (ε− 1)V arj(p†)}

=Y

2(ε− 1)V arj(p†).

Thus firms acquire information if and only if

V arj(p†)≥ cj

where cj ≡ 2Y (ε−1)

cj which is expression (11) in the text. In the text, we also denote with

lower-case letters the logs of the respective variable.Note that exactly the same calculations apply in the dynamic case, where each variable

is indexed with t. In this case the expected difference of profits at time t, in (30), is equal to

Ejt {π(P †t (j), Pt, θt)− π(Pt(j), Pt, θt)} =

Y

2(ε− 1)V arjt (p

†t)

in a second-order approximation, where the expectation operator is conditional on the privatehistory of signals including time t. So V arjt (p

†t) corresponds to the subjective contempora-

neous variance of p†t .

A.2 Proof of proposition 2

Equation (26) defines implicitly the equilibrium fraction µ as a function of r, µ(r). We havec∗ = c∗ (r, µ (r)). Differentiate implicitly (26) to get

dµ

dr= −

f(c∗)∂c∗

∂r

1 + f(c∗)∂c∗

∂µ

. (A.8)

Note that

∂c∗

∂µ=−2η2(1− r)2σ2

u(1− η)(η + (1− η)(1− r)µ

)3 (A.9)

∂c∗

∂r=

η2σ2u(

η + (1− η)(1− r)µ)3

[(1− η)(1− r)µ− η

](A.10)

As we noted in the text, when η < 1, we have strategic complementarities in information

27

acquisition, ∂c∗

∂µ< 0. The opposite happens when η > 1, ∂c∗

∂µ> 0.

The denominator of (A.8) is always positive when we have strategic substitutes. In thecase of strategic complements, we are going to assume that the complementarities are notlarge enough to make the denominator negative, so we assume that −f(c∗)∂c

∗

∂µ< 1. This

makes the LHS of (26) an increasing function of µ with slope less than unity. This is whatwe call a “stable” equilibrium.

The above discussion implies that the sign of dc∗/dr is determined by the sign of ∂c∗/∂r.The result follows from (A.10).

B Infinite-horizon model

B.1 Proof of proposition 3

Derivations of (34) and (35). We proceed using the method developed in Woodford(2002). We claim that the relevant hidden state is

Xt =

[p∗tpt

]and guess that it evolves according to a linear law of motion

Xt = f +MXt−1 +mut, (B.1)

where

f ≡[p∗

p

], M ≡

[ρ 0a b

], m ≡

[1c

],

are vectors and matrices to be determined. Note that our variables of interest are the pricesof the informed firms which can be written as p† = η′Xt and that of the uninformed whichcan be written as pt (i) = η′Ei

tXt, where η′ = (η, 1− η).Let e1 = (1, 0)′. We can write the following system

Xt = f +MXt−1 +mut

sit = e′1Xt + ξjt

where the second line corresponds to the observational equation. We proceed assuminga stationary filtering problem. The filtering equation of a generic uninformed firm j is givenby

EjtXt = Ej

t−1Xt +K(sjt − Ejt−1p

∗t ), (B.2)

where K is the vector of Kalman gains pre-multiplied with M−1. Using (B.1) we obtain thatEjt−1Xt = f +MEj

t−1Xt−1 and Ejt−1p

∗t = p∗ + ρEj

t−1p∗t−1; we can then write (B.2) as

EjtXt = f +MEj

t−1Xt−1 +K(sjt − p∗ − ρEjt−1p

∗t−1).

Aggregating among all agents j that are uninformed and guessing that in equilibrium µ will

28

be non-random (as in the static case) we obtain

EtXt = f +MEt−1Xt−1 +K(p∗t − p∗ − ρEt−1p

∗t−1

),

= f +MEt−1Xt−1 + ρK(p∗t−1 − Et−1p

∗t−1

)+Kut, (B.3)

which is the law of motion of the average estimate, where we have used the law of largenumbers.

Our target is to express the price level pt in terms of Xt−1. The general price index canbe expressed as a function of the full information price and average expectations as

pt = δ[(1− µ) p∗t + µEtp

∗t

]+ (1− δ) Etpt

= (δµ, 1− δ) EtXt + δ (1− µ) p∗t= (δµ, 1− δ) EtXt + δ (1− µ)

(p∗ + ρp∗t−1 + ut

).

Using the law of motion (B.3) to substitute for EtXt and collecting terms we have

pt = [(δµ, 1− δ) f + δ (1− µ) p∗] + ρ[δ (1− µ) + K

]p∗t−1 + (δµ, 1− δ)MEt−1Xt−1

−ρKEt−1p∗t−1 +

[δ (1− µ) + K

]ut,

where K ≡ (δµ, 1− δ)K. Finally, using the definition of M and Et−1Xt−1 and noting that

(δµ, 1− δ) f + δ (1− µ) p∗ = δp∗ + (1− δ) p

we obtain that

pt = δp∗ + (1− δ) p+ ρ[δ (1− µ) + K

]p∗t−1 +

[δµρ+ (1− δ) a− ρK

]Et−1p

∗t−1

+ (1− δ) bEt−1pt−1 +[δ (1− µ) + K

]ut. (B.4)

Sincept−1 = δµEt−1p

∗t−1 + (1− δ) Et−1pt−1 + δ (1− µ) p∗t−1,

we can use this expression to substitute for Et−1pt−1 in (B.4) and arrive at

pt = δp∗ + (1− δ) p+[δ (1− µ) (ρ− b) + ρK

]p∗t−1

+[δµ (ρ− b) + (1− δ) a− ρK

]Et−1p

∗t−1

+bpt−1 +[δ (1− µ) + K

]ut. (B.5)

We note that (B.1) impliespt = p+ ap∗t−1 + bpt−1 + cut. (B.6)

We can then match the coefficients between (B.5) and (B.6) and obtain

δp∗ + (1− δ) p = p

δ (1− µ) (ρ− b) + ρK = a

δµ (ρ− b) + (1− δ) a− ρK = 0

δ (1− µ) + K = c.

29

Solving this system and using the definition of K = (δµ, 1− δ)K, we get

p = p∗

a =ρ

δµ+ 1− δK = ρη′K = ρk

since η = δµ/(δµ+ 1− δ) and k ≡ η′K. Moreover

b = ρ− a = ρ(

1− k)

andc = (δµ+ 1− δ) k + δ (1− µ) .

The vector of (pre-multiplied) Kalman gains satisfies the equation

K = Σe1

(e′1Σe1 + σ2

ξ

)−1, (B.7)

where Σ is the variance of the one step ahead forecast error which satisfies the followingstationary version of the Riccatti equation

Σ = MΣM ′ +mm′σ2u −

(e′1Σe1 + σ2

ξ

)−1MΣe1e

′1ΣM ′. (B.8)

Thus in our guess-and-verify approach we expressed M and m as a function of k whichdepends on the vector of Kalman gains K which in turn depends on Σ. But Σ depends onM and m by (B.8). So it remains to solve for this fixed point. Let

Σ =

[σ11 σ12

σ12 σ22

].

Solving the upper left block of the Riccatti equation (B.8) we find that σ11 satisfies thequadratic

σ211 +

[(1− ρ2

)σ2ξ − σ2

u

]σ11 − σ2

ξσ2u = 0.

The positive root (since σ11 is a variance) of this quadratic is

σ11 =1

2σ2u

{1−

(1− ρ2

)λ+

√(1− (1− ρ2) λ

)2+ 4λ

}where λ ≡ σ2

ξ

σ2u. From the lower left block of the Riccatti we derive

σ12 =ρ2kσ11σ

2ξ + (σ11 + σ2

ξ )[(δµ+ 1− δ) kσ2

u + δ (1− µ)σ2u

][1− ρ2(1− k)

]σ2ξ + σ11

30

and -using (B.7)- we obtain

k = η′K = ησ11

σ11 + σ2ξ

+ (1− η)σ12

σ11 + σ2ξ

.

This is a system of two equations in the two unknowns (σ12, k). Solving the system and usingour solution for σ11 we finally arrive at the quadratic expression for k

Q(k) = ρ2λk2 +[λ(1− ρ2

)+ δ]k − δ = 0. (B.9)

At first, note that the k that solves the quadratic for ρ = 0 agrees -as expected- with thestatic case. Proceeding with the case of ρ 6= 0, we note that the discriminant of (B.9) ispositive, so there are two real roots. Furthermore, since Q (0) < 0 and Q (1) = λ > 0, one isnegative and the other positive and less than unity. Note that subtracting pt from p∗t we getan expression for the output deviation

yt = ρ(1− k)yt−1 + (1− c)ut,

since p∗t +ln Y = lnMt = pt+lnYt. In order to have a stationary solution for output we need∣∣∣1− k∣∣∣ < |ρ|−1. It follows that k should satisfy the restriction 1− ρ−1 < k < 1 + ρ−1. Since

Q (1− ρ−1) = −ρ−1((ρ− 1)2 λ+ δ

)< 0, only the positive root of the quadratic satisfies the

restriction. Thus

k =1

2ρ2

ρ2 − 1− δ

λ+

√[(1− ρ2) +

δ

λ

]2

+ 4ρ2δ

λ

.

Having solved for the laws of motion of Xt and EjtXt, we can derive the laws of motion

of the prices of interest p† = η′Xt and pt (j) = η′EjtXt and obtain expressions (34) and (35)

in the main text, respectively.

Derivation of (36). Defining qjt ≡ p†t − pt (j) = p†t − Ejt p†t we obtain

qjt = ρ(1− k)qjt−1 + δ(1− k)ut − kξjt . (B.10)

At first notice that Ejt qjt = Ej

t (p†t − pt(j)) = 0 and Ej

t−1qjt = Ej

t−1(p†t − pt(j)) = Ejt−1p

†t −

Ejt−1E

jt p†t = 0. Calculating variances conditional on the private history until last period we

getV arjt−1(qjt ) = ρ2(1− k)2V arjt−1(qjt−1) + δ2(1− k)2σ2

u + k2σ2ξ .

Note that V arjt−1(qjt−1) = Ejt−1(qjt−1)2 = V arjt−1(p†t−1). Moreover, V arjt−1(qjt ) = Ej

t−1(qjt )2 =

Ejt−1{E

jt (q

jt )

2} = Ejt−1V ar

jt (p†t) = V arjt (p

†t), where the last step follows from the non-

randomness of the variances of the filter. The expression in the text for the contempora-neous variance follows by using the stationarity of the filter and the fact that (B.9) impliesδ(1− k) = λk[1− ρ2(1− k)].

31

B.2 Proof of proposition 4

We will now proceed to derive the condition for excess volatility of the prices of the uninformedfirms. Taking unconditional variances in (B.10) we obtain that

var(qjt ) =1

1− ρ2(1− k)2[δ2(1− k)2σ2

u + k2σ2ξ ].

Furthermore note that, since

var(qjt ) = var(p†t) + var(pt(j))− 2cov(p†t , pt(j))

andcov(p†t , pt(j)) = cov(qjt , pt(j)) + var(pt (j)),

we havevar(p†t) = var(qjt ) + 2cov(qjt ,pt(j)) + var(pt(j)).

Dividing over var(p†t) we obtain

var(pt(j))

var(p†t)= 1− var(qjt ) + 2cov(qjt , pt (j))

var(p†t).

So the ratio can exceed unity only if I ≡ var(qjt ) + 2cov(qjt , pt (j)) < 0. Note that

cov(qjt , pt (j)) =1

1− ρ2(1− k)2[ρ2(1− k)k · cov(p∗t−1, q

jt−1) + δ(1− k)kσ2

u − k2σ2ξ ].

Using the law of motion for the full information price and qjt we derive that

cov(p∗t , qjt ) =

δ(1− k)σ2u

1− ρ2(1− k)

and plugging it in the previous expression we finally obtain

cov(qjt , pt(j)) =1

1− ρ2(1− k)2

[δ(1− k)k

1− ρ2(1− k)σ2u − k2σ2

ξ

].

Therefore

I =k2

1− ρ2(1− k)2

[δ(1− k)

k

(δ(1− k)

k+

2

1− ρ2(1− k)

)− λ

]σ2u.

In order to have I < 0, we need

δ(1− k)

k

[δ(1− k)

k+

2

1− ρ2(1− k)

]− λ < 0.

32

Using as before the fact that δ(1− k) = λk[1− ρ2(1− k)] we derive condition (38).

C Information costs

We convert the lifetime costs to costs per period by multiplying it by the factor (1− β). Thesteady state real profits are π = Y /ε. Recall that

cj =2

(ε− 1)

cj

Y.

Then the equilibrium fraction of uninformed agents is given by

µ = 1− Pr(cj ≤ c∗).

where, Pr refers to the probability measure of costs. In particular

Pr(cj ≤ c∗) = Pr(cj (1− β) ≤ c∗ (1− β) = V arjt (p†t))

and

Pr

(cj (1− β)

Y≤ (ε− 1)

2V arjt (p

†t)

)= Pr

(cj (1− β)

π≤ ε (ε− 1)

2V arjt (p

†t)

).

Given our assumption of a uniform distribution for the costs as a fraction of steady state realprofits with minimum zero and mean 4.61%, it follows that

µ = 1− U(ε (ε− 1)

2V arjt (p

†t)

),

where U(·) is the corresponding c.d.f. This equation gives the implicit equation (41).

33

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